When do you add exponents and when do you multiply them?
Updated
The short answer
When you multiply powers of the same base, add the exponents, so x² · x³ = x⁵. When you raise a power to a power, multiply them, so (x²)³ = x⁶, and writing x² · x³ = x⁶ mixes up the two rules.
What it looks like
| Wrong | Right |
|---|---|
| x² · x³ = x⁶ | x² · x³ = x⁵ |
| (x²)³ = x⁵ | (x²)³ = x⁶ |
| (2x)³ = 2x³ | (2x)³ = 8x³ |
A number shows the first one quickly. With x = 2, 2² · 2³ = 4 × 8 = 32, which is 2⁵, while 2⁶ = 64.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
The two rules look alike, and both examples use a 2 and a 3, so it's easy to grab the wrong one. The way out is to remember what an exponent counts: how many times the base is multiplied.
- x² · x³ is (x · x)(x · x · x). That's five x's, so x⁵.
- (x²)³ is x² three times, (x · x)(x · x)(x · x). That's six x's, so x⁶.
- x⁵/x² is five x's over two x's. The two on the bottom cancel two on top, so x³ is left.
Adding is a different story. x² + x³ can't be written as one power, because x² and x³ aren't like terms. With x = 2 it's 4 + 8 = 12, which isn't a power of 2 at all.
The last row of the table is a different slip, about what the exponent applies to. An exponent belongs to whatever sits directly to its left. In 2x³ only the x is cubed. In (2x)³ everything inside the parentheses is cubed, so the 2 is cubed too: (2x)(2x)(2x) = 8x³. Paul Dawkins's notes on exponents list forgetting the exponent on the number in front as one of the more common mistakes. The same idea explains why −3² = −9 while (−3)² = 9.
How to catch it yourself
When you're not sure which rule applies, write the powers out as repeated multiplication and count. It only takes a moment, and counting can't mix up the rules. Then point at the exponent in your answer and ask yourself:
How many x's are multiplied together in x² · x³?
When parentheses have an exponent outside, check that every factor inside got the exponent, numbers included. Putting in x = 2 works as a check for any of these rules: (2²)³ = 4³ = 64, which is 2⁶.
Try these
Simplify each one.
- a⁴ · a³
- (y³)⁴
- (3x²)²
Answers
- a⁷
- y¹²
- 9x⁴
How alai helps
In Homework help, when you pause after writing a step, Ally reads it and asks about the first line that doesn't follow from the one before. The hint stops one step short, so the fix is still yours to make. Ally also remembers slips from past sessions, so a later session can start from there.